Question

the table shows the number of visitors to a web page over a period of several months. write an explicit formula and a recursive formula to model the data.
month | visitors
1 | 15
2 | 30
3 | 60
4 | 120
5 | 240

the explicit formula is $a_n = \square$

Explanation:

Step1: Identify the sequence type

Looking at the number of visitors: 15, 30, 60, 120, 240... We can see that each term is multiplied by 2 to get the next term. So this is a geometric sequence.

Step2: Recall the explicit formula for a geometric sequence

The explicit formula for a geometric sequence is \( a_n = a_1 \times r^{n - 1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step3: Determine \( a_1 \) and \( r \)

Here, \( a_1 = 15 \) (the number of visitors in the first month) and \( r = 2 \) (since \( \frac{30}{15}=2 \), \( \frac{60}{30}=2 \), etc.).

Step4: Substitute \( a_1 \) and \( r \) into the formula

Substituting \( a_1 = 15 \) and \( r = 2 \) into \( a_n = a_1 \times r^{n - 1} \), we get \( a_n = 15 \times 2^{n - 1} \).

Answer:

\( 15 \times 2^{n - 1} \)